An Advanced Tropospheric Delay Model Based on Gaussian Mixed Long Short-Term Memory Network

In spaceborne microwave remote sensing and geodesy, tropospheric delay has emerged as a critical factor affecting the precision of measurements. While the Global Navigation Satellite System (GNSS) offers reliable station-wise zenith total delay (ZTD) products, their spatial resolution is inherently constrained by the GNSS station distribution. Conversely, empirical models combined with numerical weather models (NWMs), such as the fifth generation of European Reanalysis (ERA5) and Vienna Mapping Functions 3 (VMF3), can generate global gridded ZTD estimates. Yet, they exhibit centimeter-level discrepancies when benchmarked against GNSS-derived ZTD. This article proposes a deep learning method based on the Gaussian mixture long short-term memory (GM-LSTM) network, which learns the mapping of probability density between ZTD derived from the empirical model to the ZTD derived from GNSS. Once this mapping is learned, it can be used to infer the ZTD probability distribution and its uncertainty at any location within the study area. Upon evaluation across eight different latitude regions in Europe, the ZTD inferred by the proposed GM-LSTM model reaches the state-of-the-art level with an average root-mean-square error (RMSE) of 4.6 mm. Compared with ZTD estimated from deep neural network (DNN), ERA5 ray tracing, VMF3, and the Generic Atmospheric Correction Online Service (GACOS), the proposed GM-LSTM model achieved average performance improvements of 41.78%, 68.20%, 49.56%, and 50.43%, respectively. Verified by the meteorological records, the proposed GM-LSTM model can effectively reflect the uncertainty caused by spatially heterogeneous rainfall events. With homogeneous training data, it shows good performance in heavy rainfall, which is not matched by other ZTD estimation methods.